The invention relates to a method and apparatus for the structural analysis of components, of particular but by no means exclusive application in determining the deformation and stress distribution within an object that is subject to loads, especially in the analysis of injection molded parts to determine their deformation and stress levels under external or internal loading.
This invention is of most particular application in the structural analysis of thin walled structures, the most important geometric feature of which is that one dimension, the thickness, is at least several times smaller than the other two dimensions. Examples of such structures are injection-molded parts of metal, ceramic or polymeric material, metal castings, and structures formed from sheet metal.
Manufacturers of components and parts, in particular by injection molding, prefer to move structural analysis of the parts upstream in the design process in order to reduce design costs and time-to-market. In order to satisfy the demands of design engineers, existing products integrate finite element analysis (FEA) and Computer Aided Drafting (CAD). Pro-Engineer (trade mark), CATIA (trade mark), I-DEAS (trade mark), Solid Works (trade mark) and Solid Edge (trade mark) brand solid modelling packages are commonly used in mechanical design and drafting. These packages may be used to generate three dimensional, photo-realistic descriptions (known as xe2x80x98solid modelsxe2x80x99) of the component geometry. At present, the structural analysis packages directly based on solid models use solid elements such as tetrahedra and hexahedra.
For structural analysis of solid models, the region defined by the solid model is divided into a plurality of small elements called solid elements. This process is called meshing and the resulting collection of solid elements is called a solid mesh. Solid elements are usually simple geometric solids such as tetrahedra or hexahedra. Generation of the solid mesh has been improved in recent years though for complex parts, it is rarely automatic. Frequently the user will need to remove features from the solid model to allow the mesh to be generated successfully. This can be very time consuming and in extreme cases may necessitate remodelling of the component or some region of the component.
The use of solid elements has no theoretical advantage over the use of shell elements for thin walled structures, at least in determining the structural response of the component under load. However, the majority of component modelling is done in solid modelling systems, so the use of solid elements is more natural and allows a better interface between the geometric solid model and the mesh used for analysis. A particular problem arises with components that are thin walled. In this case, to achieve accurate results, it has been necessary to ensure that there are several well-shaped solid elements in the thickness direction. This leads to a large number of elements in the model and hence long computing times and large memory requirements. While it may be possible to use a higher order element to reduce the number of elements through the thickness, the automatic generation of such a mesh is still difficult. To reduce the size of large solid element models, the user may increase the characteristic element dimension and remesh the geometry. The automatic mesh generator will then generate fewer elements but the resulting finite element mesh may not be able to model the real stress distribution, owing to too few elements. Moreover a solid element mesh with an insufficient number of elements through the thickness has other problems, such as ill-conditioned stiffness matrix, shear locking and poor simulation of pure-bending and bending-dominated structural response. These can seriously affect the reliability of finite element analysis.
Thin walled structures typically consist of plate and shell components. There exist several classical theories for plates and shells. Particularly well-known are the Kirchhoff theory and Mindlin-Reissner theory. In the Kirchhoff theory, it is assumed that normals to the mid-surface before deformation remain straight and normal to the mid-surface after deformation. The Mindlin-Reissner theory employs the hypothesis that normals to the mid-surfaces before deformation remain straight but not necessarily normal to the plate after deformation. The stress normal to the midsurface is disregarded in both theories. Many kinds of plate and shell elements have been established based on the different plate and shell theories over the past 35 years. These permit accurate finite element analysis of thin walled structures but require a model which must be derived from the solid geometry in the CAD system. A shell element model for analysis consists of a lattice of planar or curved shell elements. Generally the shape of the elements are of simple geometric shape such as triangles or quadrilaterals. The element thickness is not explicitly shown on the element, though it is a property of the element. A shell element model may be generated from a solid geometry by forming a mesh of shell elements on the imaginary surface lying between the outer walls of the solid model. This surface is frequently called the mid-plane surface of the solid model. It is not possible to define the mid-plane surface automatically in all cases, so the generation of a shell element model is frequently a laborious task involving the construction of a separate model for analysis.
Thus, the solid element approach to the structural analysis of a thin walled. component has the advantage of easy interfacing to the solid geometry, while the shell element approach has the advantages of good structural performance, low compute times, low memory requirements and ease of mesh generation. However, the solid element approach has the disadvantages of difficult mesh generation, high element number, long compute times, high memory requirements and poor results if insufficient elements through thickness for low order elements, while with the shell element approach it is difficult to derive a mid-plane for creating a shell mesh.
Existing boundary element methods permit structural analysis of components by using a mesh generated on the surface of the solid geometry, but traditional boundary element methods require that the material be isotropic and linear. Boundary element methods also lead to large unbanded systems of equations, the solution of which requires large amounts of memory.
As described above, the shell element is appropriate for the structural analysis of (generally thin walled) structures if the mid-plane model is available. Well-established plate-shell theories are used in the shell element so that the number of dimensions is reduced sensibly from three to two, i.e. from a solid to a surface. On the other hand, it is desirable to directly use the solid model from a CAD package for finite element analysis.
Such shell elements are generally triangular or quadrilateral in shape, and may be planar or curved. At each node there are 5 or 6 degrees of freedom (dof). The degrees of freedom, in the most general case, comprise three translations and three rotations. FIG. 1 shows a triangular shell element which has a local coordinate system attached to it; the degrees of freedom are referenced to this coordinate system. Translational degrees of freedom for node n (n=1,2 or 3) in the local x, y and z directions are denoted by uxn, uyn and uzn respectively. Similarly rotations about the local x, y and z axes are denoted by xcex8xn, xcex8yn and xcex8zn respectively. The surface through the element on which the nodes are located is called the reference surface. Usually a shell element is formulated with the midsurface as the reference surface. If the element reference surface is not on the midsurface, the element is said to be an eccentric shell element, with the distance by which the reference surface is displaced from the midsurface termed the eccentricity, xcex5 (see FIG. 2), in which is also indicated the reference surface 10, the midsurface 12 and nodes 14). The formulation for an eccentric shell element can be established by extending the formulation of the normal shell element. The relationship between either the strains on the midsurface and reference surface or the degrees of freedom of nodes on the midsurface and reference surface can be used for extension from the formulations of normal shell elements to the formulations of eccentric shell elements.
FIGS. 3A, 3B and 3C show three planar triangular shell elements of thickness t with the reference surface placed at three possible locations: the bottom, midsurface and top of the element respectively. In these figures, xe2x80xa2 again denotes node position.
It is an object of the present invention to address the limitations of using solid and shell elements for the structural analysis of thin walled structures while retaining at least some of the advantages of shell elements for thin walled structures.
According to the present invention, therefore, there is provided a method for analysing the structural response of an object having an outer surface comprising a plurality of surface portions, involving:
forming a three dimensional model of said object, said model comprising a surface mesh representative of said outer surface and comprising a plurality of eccentric shell elements, wherein each of said elements is defined by a plurality of nodes on said surface and each of said nodes has one or more degrees of freedom;
assigning to each of said elements a thickness indicative of half of the thickness of said object at said respective element;
defining, for each of said elements, a reference surface that includes the nodes of said respective element and that is coincident with said surface mesh at said respective element;
for a pair of opposed portions of said surface, establishing a constraining relationship between said degrees of freedom of each of said nodes on the first of said opposed portions and said degrees of freedom of one or more of the nodes on the second of said opposed portions;
performing a finite element structural analysis of said object; and
outputting one or more results of said structural analysis.
Thus, according to the invention the reference surfaces of the shells are offset and multipoint constraints (between nodes) are used and thicknesses assigned to ensure that the collection of elements on corresponding surface portions have the same structural characteristics as a mesh of shell elements located at the midplane of the solid model and having the local thickness of the solid model at that location. Performing the finite element structural analysis includes defining boundary conditions and external and/or internal loading. Thus, the present invention uses modified shell elements defined on the surface mesh to perform structural analysis of the solid component.
The results of the analysis may include data or images indicating the expected deformation and stress state of the object under some internal or external load.
The portions of the surface may not be planar.
Preferably, for each of said nodes of said first of said opposed portions, the method includes establishing a constraining relationship between said degrees of freedom of said node and the degrees of freedom of an opposite node, being that node on the second of said pair of opposed portions opposite said node, when said opposite node exists, or between said degrees of freedom of said node and said degrees of freedom of the nodes of an opposite element, being that element of said opposed portion opposite said node, when opposite node does not exist.
Portions of the surface may be classified as sheets or edges, and the elements accordingly as sheet elements or edge elements.
Preferably the thickness indicative of half of the thickness of said object is derived from the actual thickness of said object if it is possible to define such a thickness. Where it is not possible to define the thickness of said object, the thickness of said elements may be taken to be the thickness of adjacent elements, or proportional to the thickness of adjacent elements.
Thus, even where it may be impossible to define a meaningful thickness, such as at the edges of the object or at its ends, a thickness indicative of the thickness can still be defined.
Preferably the method includes selecting said pair of opposed portions to be those opposed portions of said surface of said object between which said object is thinnest.
Thus, the method is expected to provide optimal results for thin objects, in which case the constraining relationship is preferably defined between the nodes of the surface portions between which said object is generally thinnest.
The mesh may comprise a lattice of triangles, quadrilaterals or other simple shapes (including polygons), any of which may be planar or curved. Many of these shapes can be readily generated by CAD systems.
The structural analysis may be of the effects of internal stresses or loads induced in the object during its manufacture, in which case the structural analysis will be of the resulting deformation or warping of the object caused by these internal loads. Such deformation is referred to below as xe2x80x98warpagexe2x80x99.
The method may create the surface mesh by creating or importing a stereolithography representation of the object. More preferably, the method may then include improving the stereolithography representation (such as by generating a finer mesh with smaller elements).
The present invention also provides an apparatus for analysing the structure of an object having an outer surface comprising a plurality of surface portions, having:
modelling means for forming a three dimensional model of said object, said model comprising a surface mesh representative of said outer surface and comprising a plurality of eccentric shell elements, wherein each of said elements is defined by a plurality of nodes on said surface and each of said nodes has one or more degrees of freedom;
means for assigning to each of said elements a thickness indicative of half of the thickness of said object at said respective element;
means for defining, for each of said elements, a reference surface that includes the nodes of said respective element and that is coincident with said surface mesh at said respective element;
constraining means for establishing, for a pair of opposed portions of said surface, a constraining relationship between said degrees of freedom of each of said nodes on the first of said opposed portions and said degrees of freedom of one or more of the nodes on the second of said opposed portions;
analysis means for performing a finite element structural analysis of said object; and
outputting means for outputting one or more results of said structural analysis.
Each of these means may include computer program products or portions, and the apparatus preferably includes a computer for executing such computer program portions.
The results of the analysis may include data or images (in the form of, for example, printouts, displays or computer files) indicating the expected deformation and stress state of the object under some internal or external load.
Preferably said constraining means is operable, for each of said nodes of said first of said opposed portions, to establish a constraining relationship between said degrees of freedom of said node and the degrees of freedom of an opposite node, being that node on the second of said pair of opposed portions opposite said node, when said opposite node exists, or between said degrees of freedom of said node and said degrees of freedom of the nodes of an opposite element, being that element of said opposed portion opposite said node, when opposite node does not exist.
Preferably said means for assigning to each of said elements a thickness (i.e. indicative of half of the thickness of the object at the respective element) is operable to employ a thickness indicative of the total thickness of said object at said respective element the actual total thickness of said object if it is possible to define such a thickness, andxe2x80x94where it is not possible to define the thickness of said objectxe2x80x94to assign to said elements a thickness equal to the thickness of adjacent elements, or proportional to the thickness of adjacent elements.
Thus, in calculating the thickness indicative of half of the thickness of the object, the apparatus preferably derives the half thickness from an actual total thickness, if possible.
Preferably the apparatus is configured to select as said pair of opposed portions those opposed portions of said surface of said object between which said object is thinnest.
The mesh may comprise a lattice of planar or curved triangles, quadrilaterals or other simple shapes. Many of these shapes can be readily generated by CAD systems.
The analysis means may be operable or configured to analyse the effects of internal stresses or loads induced in the object during its manufacture, in which case the structural analysis will be of the resulting deformation or warping of the object caused by these internal loads.